Sketching Rational Functions

Transcription

1 00 D.W.MacLean: Graphs of Rational Functions-1 Sketching Rational Functions Recall that a rational function f) is the quotient of two polynomials: f) = p). Things would be simpler q) if we could assume that p and q had no common roots, but we cannot. The roots of the numerator p) are the possible roots of f, and the roots of the denominator give us the possible locations of the vertical asymptotes of f. From the Quotient Rule f ) = q)p ) p)q ) q)) = qp pq q) we know that the denominator of the first derivative is the square of the denominator of f, and the denominator of the second derivative is the fourth power of the denominator of f, so they contain no new information: The asymptotes are at the roots of q) only. The additional information about critical and inflection points comes from the numerators of the first and second derivatives respectively. The general formula for the second derivative is: f = q) [ qp pq ] q) ) qp pq ) q) ) = q [ p q pq ] qp q + pq ) The numerator of the second derivative is seldom easy to work with! Clearly the sketching of a rational function is made easier if we have available the graphs of the numerator and denominator. In our eamples we will first graph a specific function, and then look at the general class of functions of which it is a special case. Note: For most problems, the graph can be viewed interactively using Java applets with Netscape Communicator or Internet Eplorer. Colour Conventions: The graph of the function being studied is drawn in black, that of the first derivative is in red, and that of the second derivative is in blue. The numerator s graph is drawn in orange. The denominator s graph is drawn in purple. The asymptotes are drawn in green, ecept when they coincide with the aes, which are black. q

2 00 D.W.MacLean: Graphs of Rational Functions- Eample 1: The graph of f) = is: Step 1: f ) = 1) + 1) + 1) 1) 1) = and f ) = ) ) 1) 4 = 1). The only interesting value of f is 1. 1)1) + 1)1) 1) = = 1) 1) Step : Put these values of into increasing order. 1. Step : Put together as good a table as you can showing the signs of f ) and f ) on the intervals into which the interesting values divide the domain of f., 1) 1 1, ) f ) UND + f ) UND f) UND Step 4: Plot the interesting points and connect them with curves which are either left or right half-smiles or half-frowns. But now we have a problem: the only interesting -value, 1, doesn t have a corresponding y-value because 1 is not in the domain of f! Even though the function is not defined at 1, we can compute the left- and right- hand limits there:

3 00 D.W.MacLean: Graphs of Rational Functions- + 1 lim f) = lim =, and lim f) = lim =+. We can also compute the limits at and + : + 1 lim f) = lim 1 = 1, and lim + 1 f) = lim = 1. We can also easily see that f 1) = 0. We can represent these facts in a modified table, using a bit of shorthand notation:, 1) , ) f ) f ) 0 0 f) 1 f 1) = 0 1 We use this to put together a sketch of the graph: The horizontal line y = 1 and the vertical line = 1 are called asymptotes of the graph.

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5 00 D.W.MacLean: Graphs of Rational Functions-5 Eample : The graph of f) = a + b, where ad bc 0, is: c + d Step 1: f ) = c + d)a + b) a + b)c + d) c + d) ad bc = 0 = c + d)a) a + b)c) c + d) = ad bc c + d) 0 unless and f ad bc ) = c c + d) 4. The only interesting value of f is d, where we have a vertical asymptote. c Step : Put these values of into increasing order. d c. Step : Put together as good a table as you can showing the signs of f ) and f ) on the intervals into which the interesting values divide the domain of f., d ) c d c dc, ) f ) 0 sign cad bc)) UND sign cad bc)) 0 f ) 0 sign ad bc) UND sign ad bc) 0 a f) f b ) a = 0 UND c a c Step 4: Plot the interesting points and connect them with curves which are either left or right half-smiles or half-frowns.

6 00 D.W.MacLean: Graphs of Rational Functions-6 Again we have a problem: the only interesting -value, d c, doesn t have a corresponding y-value because d c is not in the domain of f! Even though the function is not defined at d, we can compute the left- and right- hand limits there: c lim f) = d c lim d c a + b c + d =, and lim d c We can also compute the limits at and + : + a + b f) = lim d + c + d =+. c a + b lim f) = lim c + d = a c, and lim a + b f) = lim + + c + d = a c. We can also easily see that f b ) = 0. We can represent these facts in a modified table, using a bit of a shorthand notation:, d ) c d c d c + d c, ) f ) f ) 0 0 a f) f b ) = 0 sign a c a c ) sign a c ) a c We use this to put together a sketch of the graph: The horizontal line y = a c and the vertical line = b a are called asymptotes of the graph.

7 00 D.W.MacLean: Graphs of Rational Functions-7 We sketch the graph with a = 1, b =, c =, and d =. y

8 00 D.W.MacLean: Graphs of Rational Functions-8 Eample The graph of f) = Step 1: 1 is: f ) = 1)) ) 1) 1) = 1)1) )) 1) = + 1 1) and f ) = 1) + 1) + 1) 1) ) 1) ) = 1) ) + 1) 1)) ) 1) 4 = 1) + 1)) 1) = 1 + 1) = = 1) 1) The only interesting values of f are 1, 0, and 1. Step : Put these values of into increasing order. 1, 0, 1. Step : Put together as good a table as you can showing the signs of f ) and f ) on the intervals into which the interesting values divide the domain of f., 1) , 0) 0 0, 1) , ) f ) f ) f)

9 00 D.W.MacLean: Graphs of Rational Functions-9 Step 4: Plot the interesting points and connect them with curves which are either left or right half-smiles or half-frowns. We use this to put together a sketch of the graph: The horizontal line y = 0 and the vertical lines = 1 and = 1 are the asymptotes of the graph. y

10 00 D.W.MacLean: Graphs of Rational Functions-10 Eample 4 The graph of f) = Step 1: a b) c) with b c f ) = b) c) a) a) b) c)) b) c)1) a) b + c)) b) c)) = b) c) = b + c) + bc a + b + c) + ab + c)) b) c) = b + c) + bc + a + b + c) ab + c)) b) c) = a + ab + c) bc b) c). The discriminant of the numerator is [ ) a) 41)[ab + c) bc] = 4a 4[ab + c) bc] = 4[a ab + c) + bc] = 4 a b+c ) ] b c by completion of squares), so f ) will have real roots whenever the distance of a from b + c is greater than or equal to one-half the distance from b to c. This implies that a is not in the closed interval from b to c. [ ) a ± 4 a b+c ) ] b c These roots are = = a ± a b + c ) ) b c. Computation and analysis of f ) is left as an eercise for those with greater patience than the author! Some of the interesting values of f are b, c, and possibly a ± numbers. a b + c ) ) b c if they are real The horizontal line y = 0 and the vertical lines = b and = c are the asymptotes of the graph.

11 00 D.W.MacLean: Graphs of Rational Functions-11 We sketch the graph with a = 1, b = 0, c =. The reader is encouraged to eperiment with the Java applet. y

12 00 D.W.MacLean: Graphs of Rational Functions-1 Eample 5 y = f) = 1) Solution: Clearly Df ) =, 1) 1, ), Zf ) ={0}, and Uf ) ={1} f + 1) ) = 1),soZf ) ={ 1}, Uf ) ={1}, f + ) ) = 1) 4,soZf ) ={ }, Uf ) ={1}, so the important -values are, 1, and 1: If ) ={, 1, 1}. The -ais is a horizontal asymptote, and the line = 1 is a vertical asymptote. There is a relative minimum at = 1, and an inflection point at =. We construct a table showing the signs of the first and second derivatives, where the entries 1 and 1 + in the row are abbreviations for lim and lim : 1 1 +, ), 1) 1 1, 1) , ) f ) f ) f) f0) =

13 00 D.W.MacLean: Graphs of Rational Functions-1 Sketch: y

14 00 D.W.MacLean: Graphs of Rational Functions-14 Eample 6 y = f) = a b), a<b Solution: Clearly Df ) =,b) b, ), Zf ) ={a}, and Uf ) ={b} b) f ) a) b) ) a) ) = b) ) = b) b) a) b) 4 = + b a b) =,sozf ) ={ b + a}, Uf ) ={b}, b a) b) = f ) = b) + b a) b) ) + b a) b) ) = b) b) + b a) b) 6 = b + b a) b b + 6a + b a b) 4 = b) 4 = b) 4 =,sozf ) ={ b + a}, Uf ) ={b}, so the important -values are b, b + a, and b + a: If ) ={b, b + a, b + a}. The -ais is a horizontal asymptote, and the line = b is a vertical asymptote. There is a relative minimum at = b + a, and an inflection point at = b + a. We construct a table showing the signs of the first and second derivatives, where the entries b and b + in the row are abbreviations for lim and lim : b b +

15 00 D.W.MacLean: Graphs of Rational Functions-15 y 5, ), 1) 1 1,b) b b + b, ) f ) f ) f) f0) =

16 00 D.W.MacLean: Graphs of Rational Functions-16 Eample 7 y = f) = = = 1 + 1) 1 Solution: f ) = 6 + 1) > 0, f ) = 11 ) + 1), so the important -values are 1, 0, and 1. There are no relative etrema, but there are Inflection Points at 1 0, 1), and, 1 ). = 1 is a vertical asymptote. We construct a table showing the signs of the first and second derivatives:, 1) , 0) 0 0, 1 ) 1 1, ) f ) f ) f) f1) = 0 1

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18 00 D.W.MacLean: Graphs of Rational Functions-18 Eample 8 y = f) = a b = 1 + b a) b) 1, a b Solution: f ) = b a) b), f ) = 6b a) + b b), so the important -values are b,0,and at 0, a ), and b b, b + a b. The reader is encouraged to use the Java Applet. b. There are no relative etrema, but there are Inflection Points = b is a vertical asymptote and y = a b is a horizontal asymptote.

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20 00 D.W.MacLean: Graphs of Rational Functions-0 Eample 9 y = f) = 1 = 1 Solution: f ) = 4 + = 4, f ) = 1 5 = 6 5, so the important -values are ± 6, ±, and 0. f is an odd function, so we need only consider the graph on the interval 0, ). There is a Relative Maimum at, ), ) and a Relative Minimum at. There are Inflection 9 9 Points at 6, 5 ) 6 6, 5 ) 6 and. The aes are asymptotes. 6 6 We construct a table showing the signs of the first and second derivatives: 0 + 0, ), 6) 6 6, ) f ) f ) f) + f1) =

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